Sampling parameters of ordinary differential equations with Langevin dynamics that satisfy constraints

TL;DR

This paper introduces a Langevin dynamics-based sampling method for ODE parameters that satisfies constraints, offering a computationally efficient alternative to traditional MCMC methods, demonstrated on biochemical oscillator models.

Contribution

The paper presents a novel Langevin dynamics approach for sampling constrained ODE parameters, reducing computational costs compared to standard MCMC methods.

Findings

Achieves comparable or better performance than MCMC in sampling efficiency.

Effectively samples bifurcations and limit cycles in biochemical models.

Provides insights into speedup mechanisms through numerical experiments.

Abstract

Fitting models to data to obtain distributions of consistent parameter values is important for uncertainty quantification, model comparison, and prediction. Standard Markov chain Monte Carlo (MCMC) approaches for fitting ordinary differential equations (ODEs) to time-series data involve proposing trial parameter sets, numerically integrating the ODEs forward in time, and accepting or rejecting the trial parameter sets. When the model dynamics depend nonlinearly on the parameters, as is generally the case, trial parameter sets are often rejected, and MCMC approaches become prohibitively computationally costly to converge. Here, we build on methods for numerical continuation and trajectory optimization to introduce an approach in which we use Langevin dynamics in the joint space of variables and parameters to sample models that satisfy constraints on the dynamics. We demonstrate the method by sampling Hopf bifurcations and limit cycles of a model of a biochemical oscillator in a Bayesian framework for parameter estimation, and we attain performance that matches or exceeds the performance of leading MCMC approaches that require numerically integrating the ODEs forward in time. We describe numerical experiments that provide insight into the speedup. The method is general and can be used in any framework for parameter estimation and model selection.